Simplify the following expression and state the condition under which the simplification is valid. You can assume that $k \neq 0$. $y = \dfrac{50k + 70}{2k} \div \dfrac{10(5k + 7)}{k} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{50k + 70}{2k} \times \dfrac{k}{10(5k + 7)} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ (50k + 70) \times k } { 2k \times 10(5k + 7) } $ $ y = \dfrac {k \times 10(5k + 7)} {2k \times 10(5k + 7)} $ $ y = \dfrac{10k(5k + 7)}{20k(5k + 7)} $ We can cancel the $5k + 7$ so long as $5k + 7 \neq 0$ Therefore $k \neq -\dfrac{7}{5}$ $y = \dfrac{10k \cancel{(5k + 7})}{20k \cancel{(5k + 7)}} = \dfrac{10k}{20k} = \dfrac{1}{2} $